I've enjoyed my two trips to the Publican. The most recent (last week) more than the first (about a month after opening). The beer list is quite good with a slight emphasis on Beligians, but generally well rounded. They usually have a unique Goose Island brew on tap as well. The staff has been very helpful, and have made excellent recommendations for matching beer and food. The oysters I had on my last visit were quite possibly the best I've ever had in Chicago and the beer they paired with them (perhaps St. Feuillian Brown, but I can't recall) worked really well. The other food I've had has ranged from great (boudin noir sausage with lentils and pickled fennel; pork rinds tossed in vinegar and sea salt) to OK (the charcuterie plate I had on my first visit was a disappointment). But the menu is constantly changing, and most things sound quite good.
Other spots might be:
The Bristol (previously mentioned) - They have a nice beer list, but not nearly as extensive as The Publican or others. The food was pretty good though.
Bluebird (1749 N Damen Ave) - They have a pretty decent beer list. Their bottle list might be equal to The Publican quality-wise, but I don't recall any taps. Therefore Publican gets the edge beer-wise in my mind. Also, I felt the food and the service at the Publican was much stronger. Still, this place isn't bad.
Jerry's (1938 W Division St) - A bit of an oddball, Jerry's doesn't really fall into the same category as Publican, Bristol, or Bluebird as their food is almost strictly sandwiches rather than "finer" dining, but their beer list is just as good, if not better, than almost any place in town. If you are looking for only American craft beers, that is. And since I happen to think American craft beers are the best thing going right now, I don't complain.
I am also a fan of LTH favorite (and GNR winner) Paramount Room. Again, not "finer" dining per se, but great beer, food, and atmosphere. Another place that might fit the bill is The Gage. Less beer-centric than the other places, but I think the food is perfect upscale pub fare. I also agree that Three Floyds would be worth a trip. Another, less conventional option I tried recently was going to Mado (which is BYO) and bringing some of my own bottles of beer to pair with the food. I enjoyed it, and it was definitely more affordable.